Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,24)(20,27,28,22)(21,23)(25,31,32,29)(26,30);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(18,22,23,27)(20,28)(24,29,30,31)(25,32);(2,7)(6,14)(8,15)(9,16)(18,23)(22,27)(24,30)(29,31);(17,19,21,26)(18,24,23,30)(20,25,28,32)(22,29,27,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=hi,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389048);